Question

A particle is subjected to two simple harmonic motions in the same direction having equal amplitudes and equal frequency. If the resultant amplitude is equal to the amplitude of the individual motions. Find the phase difference between the individual motions.
a) $\dfrac{3\pi }{2}$
b) $\dfrac{2\pi }{3}$
c) $2\pi$
d) $\pi$

Answer 1

When two particles are in motion and having amplitude then resultant amplitude is calculated by resultant formula for vector addition in which two vectors representing same physical quantity act simultaneously and angle between them is called phase difference between two motions.

Complete answer:
Let us assume the amplitude of two particles executing simple harmonic motion as${{A}_{1}}\And {{A}_{2}}$.
According to question amplitude of individual motions are equal so let it be${{A}_{1}}={{A}_{2}}=A$----Eq (1)
The amplitude of resultant ${{A}_{R}}$ is equal to the amplitude of individual motion. So${{A}_{R}}=A$---Eq (2)
Let us assume the phase difference between two motions is $\phi$
According to Resultant Formula for two waves,
${{A}_{R}}=\sqrt{{{A}_{1}}^{2}+{{A}_{2}}^{2}+2{{A}_{1}}{{A}_{2}}Cos\phi }$
Put the values of ${{A}_{1}},{{A}_{2}}\And {{A}_{R}}$ from equations 1 and 2
So we get,$A=\sqrt{{{A}^{2}}+{{A}^{2}}+2{{A}^{2}}Cos\phi }$
Squaring on both sides, we get
${{A}^{2}}=2{{A}^{2}}+2{{A}^{2}}Cos\phi$

& {{A}^{2}}-2{{A}^{2}}=2{{A}^{2}}Cos\phi \\\ & -{{A}^{2}}=2{{A}^{2}}Cos\phi \\\ & Cos\phi =\dfrac{-{{A}^{2}}}{2{{A}^{2}}}=-\dfrac{1}{2} \\\ \end{aligned}$$ $$\phi ={{120}^{0}}=\dfrac{2\pi }{3}$$ So the phase difference between two motions is$$\dfrac{2\pi }{3}$$. **Hence the correct answer is option b).** **Additional Information:** The formula for resultant is $${{A}_{R}}=\sqrt{{{A}_{1}}^{2}+{{A}_{2}}^{2}+2{{A}_{1}}{{A}_{2}}Cos\phi }$$ in which if phase difference between two motions is $${{0}^{0}}$$then resultant amplitude becomes maximum. If phase difference between two motions is $${{180}^{0}}$$ then the resultant amplitude becomes minimum. **Note:** If the value of Cosine function is positive then its value lies in the first and second quadrant and if the value of cosine function is negative then by using basic laws of trigonometry we are able to find the exact value of phase difference. Here in this question the value of the cosine function is negative so it lies in the third and fourth quadrant.